Reexamination of the perfectness concept for equilibrium points in extensive games

Institute of Mathematical Economics Working Papers

23

August 1974

Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games

Reinhard Selten

IMW·Bielefeld University Postfach 100131 33501 Bielefeld·Germany email: imw@wiwi.uni-bielefeld.de

http://www.imw.uni-bielefeld.de/research/wp23.php ISSN: 0931-6558

Reinhard Selten

Reexamination of the Perfectness Concept for Equilibrium Points in Extensive

Games August 1974

Reexamination of the perfectness concept for equilibrium points in extensive games

by

Reinhard Selten

The concept of a perfect equilibrium point has been intro- duced in order to exclude the possibility that disequili- brium behavior is prescribed on unreached subgames.

(Selten 1965 and 1973). Unfortunately this definition of perfectness does not remove all difficulties which may arise with respect to unreached parts of the game. It is necessary to reexamine the problem of defining a satisfac- tory non-cooperative equilibrium concept for games in ex- tensive form. Therefore a new concept of a perfect equili- brium point will be introduced in this paper.1)

In retrospect the earlier use of the word "perfect" was premature. Therefore a perfect equilibrium point in the old Sense will be called "subgame perfect". The new definition of perfectness has the property that a perfect equilibrium point is always subgame perfect but a subgame perfect equi- librium point may not be perfect.

It will be shown that every finite extensive game with per- fect recall has at least one perfect equilibrium point.

Since subgame perfectness cannot be detected in the normal form, it is clear that for the purpose of the investiga- tion of the problem of perfectness, the normal form is

an inadequate representation of the extensive form. It will be convenient to introduce an "agent normal form" as a more adequate representation of games with perfect recall.

1) The idea to base the definition of a perfect equilibrium point on a model of slight mistakes as described in sec- tion 6 is due to John C. Harsany1. The author's earl1er unpublished attempts at a formalization of thi~ concept

were less satisfactory. I am very grateful to John C. Harsanyi who strongly influenced the content of this paper.

1. Extensive games with perfect recall

In this paper the words extensive qame will always refer to a finite game in extensive form. Agame of this kind can be described as a sextuple.

(1) r

=

(K,P,U,C,p,h)

where the constituents K,P,U,A,p and h of rare as follows:2) The game tree: The game tree K is a finite tree with a dis- tinguished vertex 0, the origin of K. The sequence of ver- tices and edges which connects 0 with a vertex x is call- ed the path to x. We say that x comes before y or that y comes after x if x is different from y and the path to y contains the path to x. An endpoint is a vertex z with the property that no vertex comes after z. The set of all end- points is denoted by Z. A path to an endpoint is called a play. The edges are also called alternatives. An alternative at x is an edge which connects x with a vertex after x.

The set of all vertices of K which are not endpoints, is denoted by X.

The player partition: The player partition P

=

(Po,...,Pn) partitions X into player

~.

Pi is called player i's player

set (Player 0 is the "random" player who represents the ran- dom mechanisms responsible for the random decisions in the game.) A player set may be empty. The player sets Pi with i = 1,...,n are called personal player sets.

The information partition: For i = 1,...,n a subset u of Pi is called eligible (as an information set) if n is not empty, if every play intersects u at most once and if the number of alternatives at x is the same for every XEU. A subset UEPO is called elegible if it contains exactly one vertex.The infor- mation partition U is a refinement of the player partition P

tnto eligible subsets u of the player sets. These sets u are called information ~.The information sets u with u~Pi are called information sets of playe~ i. The set of all information

2) The notation is different from that used by Kuhn (Kuhn 1953)

- 3 -

sets of player i is denoted by Ui. The information sets of player 1,...,n are called personal information sets.

The choice partition: For UtU let Au be the set of all alter- natives at vertices XEU. We say that a subset c of Au is

eliqible (as a choice) if it contains exactly one alterna- tive at x for every vertex XEU. The choice partition C par- titions the set of all edges of K into eligible subsets c of the Au with UtU. These sets c are called choices. The choices c which are subsets of Au are called choices

~

^u.

The set of all choices at U is denoted by Cu. A choice at a personal information set is called a personal choice. A choice which is not personal is a random choice. We say

that the vertex x comes after the choice c if one of the ed- ges in c is on the path to x. In this case we also say that c i8 on the path to x.

The probability assignement: A probability distribution Pu over Cu is called completely mixed if it assigns a positive probability pu(c) to every CtCu. The probability assign- ment p is a function which assigns a completely mixed pro- bability distribution Pu over Cu to every UtUo. (p specifies the probabilities of the random choices.)

The payoff function: the payoff function h assigns a vector h(z)

=

(h1 (z) ,...,hn(z» with real numbers as components to

every endpoint z of K. The vector h(z) is called the payoff vector at z. The component hi(z) is player i's payoff at z.

Perfeet recall: An extensive game r

=

(K,P,U,C,p,h) is called an extensive game with perfect recall if the following con- dition is satisfied for every player i

=

^{1,...,n and any two}

information sets u and v of the same player i: if one ver- tex ytV comes afte~ a choice c at u then every vertei XtV

comes after this choice c.3)

3) The concept of perfect recall has been introduced by H.W. Kuhn (Kuhn 1953)

Interpretation: In agame with perfect recall a player i who has to make adecision at one of his information sets v knows which of his other information sets have been reached by the

previous course of the play and which choices have been taken there. Obviously a player always must have this knowledge if he is a person with the ability to remember wh at he did in the past.

Since game theory is concerned with the behavior of absolute- ly rational decision makers whose capabilities of reasoning and memorizing are unlimited, agame, where the players are individuals rather than teams,must have perfect recall.

Is there any need to consider games where the players are teams rather than individuals? In the following we shall try to argue that at least as far as strictly non-cooperative game theory is concerned the answer to this question is no.

In principle it is always possible to model any given inter- personal conflict situation in such a way that every person involved is a single player. Several persons who form a team in the sense that all of them pursue the same goals can be re- garded as separate players with identical payoff functions.

Against this view one might object that a team may be united by more than accidentally identical payoffs. The team may be a preestablished coalition with special cooperative possi- bilities not open to an arbitrary collection of persons in- volved in the situation. This is not a valid objection. Games with preestablished coalitions of this kind are outside the

framework of strictly non-cooperative game theory. In a strict- ly non-cooperative game the players do not have any means of cooperation or coordination which are not explicitly modelled as parts of the extensive form. If there is something like a pre- established coalition, then the members must appear as sepa-

rate players and the special possibilities of the team must be apart of the structure of the extensive game.

In view of what has been said no room is left for strictly non-cooperative extensive games without perfect recall. In the framework of strictly non-cooperative game theory such

- 5 -

games can be rejected as misspecified models of interper- sonal conflict situations.

2. Strategies, expected payoff and normal form

In this section several definitions are introduced which refer to an extensive game r

=

(K,P,U,A,p,h).

Local strategies: A local strategy biu at the information set UEUt is a probability distribution over the set Cu of the choices at U; a probability biu(c) is assigned to every choice c at u. A local strategy biu is called pure if it assigns 1 to one choice c at u and 0 to the other choices. Wherever this can be done without danger of

confusion no distinction will be made between the choice c and the pure local strategy which assigns the probability 1 to c.

Behavior strategies: A behavior strategy bi of a personal player i is a function which assigns a local strategy biu to every UEUi. The set of all behavior strategies of

player i is denoted by Bi-

Pure strategies: A pure strategy -i of player i is a function which assigns a choice c at u (a pure local strategy) to

every UEUi. Obviously a pure strategy is a special behavior strategy. The set of all pure strategies of player i is denoted by TIi.

Mixed strategies: A mixed strategy qi of player i is a pro- bability distribution over TIi:a probability qi(~i1 is assigned to every ~iE TIi.The set of all mixed strategies qi of player i is denoted by Qi. Wherever this can be done without danger of confusion no distinction will be made between the pure strategy -i and the mixed strategy qi which assigns 1 to -i.pure strategies are regarded as

special cases of mixed strategies.

Behavior strategy mixtures: a behavior strategy mixture si for player i is a probability distribution over Bi which as- signs positive probabilities si (bi) to a finite number of ele- ments of Bi and zero probabilities to the other elements of

Bi. No distinction will be made between the behavior strategy bi and the behavior strategy mixture which assigns , to bi. The set of all behavior strategy mixtures of player i is denoted by Si. Obviously pure strategies, mixed strategies and behavior

strategies can all be regarded as special behavior strategy

mixtures.

Combinations: A combination s

=

(s

" ...,s ) of behavior Stra-

n -

~ mixtures is an n-tuple of behavior strategy mixtures

SiESi' one for each personal player. Pure strategy combinations

.

=

(."...'.n)' mixed strategy combinations and behavior strategy combinations are defined analogously.

Realization probabilities: A player i who plays a behavior strategy mixture si behaves as follows: He first employs a random mechanism which selects one of the behavior strategies bi with the probabilities si (bi). He then in the course of the play at every UEUi which is reached by the play selects one of the choices c at u with the probabilities biu(c). Let s

=

(s"...,sn) be a combination of behavior strategy mix- tures. On the assumption that the si are played by the players we can compute a realization probability p(x,s) of x under s

for every vertex XEK. This probability p(x,s) is the proba- bility that x is reached by the play, if s is played. Since these remarks make it sufficiently clear, how p(x,s) is de- fined, a more precise definition of p(x,s) will not be given here.

Expected payoffs: With the help of the realization probabili- ties an expected payoff vector H(s)

=

(H, (s),...,Hn(S» can be computed as follows:

(2) H(s)

=

~

/' p(z,s)hC:..)

L~.l ZEZ

- 7 -

Since pure strategies, mixed strategies and behavior are special cases of behavior strategy mixtures, the payoff definition (2) is applicable here, too.

strategies expected

Normal form: A normal

-

form G =(n1^,...,nniH) consists of n finite non-empty and pairwise non-intersecting pure strategy

~

ni and an expected payoff function H defined on n

=

n1x...xnn. The expected payoff function H assigns a payoff vector H(.) = (H1 (~) ,...,Hn(t»with real numbers as components to every tEn . For every extensive game r the pure strategy sets and the expected payoff function defined above generate the normal form of r.

In order to compute the expected payoff vector for a mixed strategy combination, it is sufficient to know the normal form of r. The same is not true for combinations of behavior strategies. As we shall see,in the transition from the

extensive form to the normal form some important information

is lost.

3. Kuhn's theorem

H.W. Kuhn has proved an important theorem on games with per- fect recall (Kuhn 1953, p.213). In this section Kuhn's theo- rem will be restated in a slightly changed form. For this purpose some further definitions must be introduced. As be-

fore, these definitions refer to an extensive game r=(K,p,U,A,p,h).

Notational convention: Let behavior strategy mixtures tegy mixture for player i.

The combination (s1,...,si-1' ti,si+1,...,sn) which results from s,if si is replaced by ti a~d the other components of s remain unchanged,is denoted by s/si. The same notational convention

is also applied to other types of strategy combinations.

s

=

(s1,...,sn) be a combination of and let ti be a behavior stra-

Realization equivalence: Let gy mixtures for player i. We tion equivalent if for every mixtures we have:

si and si be two behavior strate- say that si and si are realiza- combination s of behavior strategy

(3) p(x,s/si)

=

p(x,s/si) for every XEK

Payoff equivalence:

mlxtures for player valent if for every tures we have

Let si and si be two behavior strategy i. We say that si and si are payoff equi- combination s of behavior strategy mix-

(4) H(s/si) - H(s/si)

Obviously si and si are payoff equivalent if they are reali- zation equivalent, since (3) holds for the endpoints z.

Theorem 1 (Kuhn's theorem): In every extensive game with perfect recall a realization equivalent behavior strategy bi can be found for every behavior strategy mixture si of a per- sonal player i.

In order to prove this theorem we introduce some further de-

finitions.

Conditional choice probabilities: Let s

=

(s, ,...sn) be

a combination of behavior strategy mixtures and let x be a vertex in an information set u of a personal player i, such that p(x,s» o. For every choice c at u we define a conditional choice probability ~(c,x,s). The choice c contains an edge e at Xi this edge e connects x with another vertex y. The probability ~ (c,x,s) is computed as follows:

(5) ^{_ p (y,s)}

~ (c,x,s) - p (x,s)

The probability that the choice been reached.

~(c,x,s) is the conditional probability c will be taken if ~ is played and x has

- 9 -

Lemma 1: In every extensive game r (with or without per- feet reeall) on the region of those triples (e,x,s) where the eonditional ehoice probability ~(c,x,s)is defined the eonditional choiee probabilities ~(e,x,s) with XEUEUi do not depend on the eomponents s' ofJ s with i;j.

Proof: Let bi,...,b~ be the behavior strategies,which are seleeted by si with positive probabilities si(bI).For p(x,s) >0 an outside observer,who knows that e has been reaehed by the play but does not know whieh of the bi has been selected before the beginning of the game,can use this knowledge in order to eompute posterior probabi-

lities ti (bi) from the prior probabilities si (bi). T~e

posterior probability ti (bi) is proportional to si (bI) mul- tiplied by the product of all probabilities assigned

by bi to ehoiees of player i on the path to x. Obviously the ti (b~) depend on si but not on the other eomponents of s. The eonditional ehoiee probability ~(c,x,s) ean be written as follows:

k

~., ,.

.'

J J

(6) ).I (c,x,s)

= /_,

ti (bi) biu (e) j=1

This shows that p(e,x,s) does not depend on the Sj with i;j.

Lemma 2: In every extensive game r with per feet reeall, on the region of those triples (c,x,s) where the eondi- tional choice probability ).I(e,x,s) is defined, we have

(7) )j(e,x,s)

=

).I(e,y,s) for XEU and YEU

Proof: In agame with perfeet reeall for XEU,YEU and UEUi player i's ehoices on the path to x are the same ehoiees as his ehoices on the path to y. (This is not true for games without per feet reeall). Therefore at x and y the posterior probabilities for the behavior stra- tegies bi oecurring in player ils behavior strategy mixture

si are the same at both vertices. Consequently (7) fol-

lows from (6).

Proof of Kuhn's theorem: In view of lemma 1 and lemma 2 the conditional choice probabilities at the vertices x in the player set Pi of a personal player can be described by a function ~i (c,u,si) which depends on his behavior strategy mixture s1 and the information set u with XEU.

With the help of ~i(c,u,si) we aonltruct the behavior strate- gy bi whose existence is asserted by the theorem. If for

at least one s

=

(s1,...,sn) with si as component we have

~(x,s) > 0 for some XEU, we define

The construction of ry local strategies be found.

bi is completed by assigning arbitra- biu to those UEUi where no such s can

It is clear that this behavior strategy bi and the behavior strategy mixture si are realizazion equivalent.

The significance of Kuhn's theorem: The theorem shows that in the context of extensive games with perfect re- call one can restrict one's attention to behavior strate- gies. Whatever a player can achieve by a mixed strategy or a more general behavior strategy mixtures can be achiev- ed by the realization equivalent and therefore also payoff equivalent bahavior strategy whose existence is secured by the theorem.

4. Subgame perfect equilibrium points

In this section we shall introduce some further definitions which refer to an extensive game r

=

(K,P,U,A,p,h) with perfect recall. In view of Kuhn's theorem only behavior strategies are important for such games. Therefore the concepts of a best reply and an equilibrium point are formally introduced for behavior strategies only.

- " -

Best vior

reply: Let strategies best reply

b

=

(b"...,bn) be a combination of beha-

!\,

for r. A behavior strategy Di of player i to b if we have

as a

(9)

~ ~ ~

A combination of behavior strategies B

=

(o"...,Dn) is called a best reply to b if for i

=

',...,n the behavior

~

str~tegy 0i is a best reply to b.

~~librium point: A behavior

" " {e

b

=

(b"...,bn) is called is a best reply to itself.

strategy combination

{e

an equilibriumpoint if b

Remark: The concepts point can be defined mixtures. In view of

of a best reply and an equilibrium analogously for behavior strategy Kuhn's theorem it is clear that for games with perfect recall an equilibrium point in behavior strategies is a special case of an equilibrium point in be- havior strategy mixtures. The existence of an equilibrium point in behavior strategies for every extensive game with perfect recall is an immediate consequence of Kuhn's theorem together with Nash's weIl known theorem on the existence of an equilibrium point in mixed strategies for every finite game (Nash '951).

Subgame: Let r

=

(K,P,U,A,p,h) be an extensive game with or

without perfect recall. A subtree K' of K consists of a vertex x of K together with all vertices after x and all edges of K connecting vertices of K'. A subtree K' is called regular in r, if every information set in r, which contains at least one vertice of K', does not contain any vertices outside of K'. For every regular subtree K' a subgame

r' -= (K', p.',U',A',p',h')is defined as foliows: P',U',A',p' and h' are the restrictions of the partitions U,A and the functions p and h to K'.

Induced strateg1es: Let r' be a subgame of rand let

b = (b1,... ,bn) be a behav10r strategy comb1nat1on for r.

The restr1ction of b1 to the 1nformation sets of player i in r' 1s a strategy bI of player i for r'. This strategy bI

1s called 1nduced by b1 on r' and the behavior strategy combination b'

= (b;,...,b~)

defined in this way is called induced by b on r'.

Subgame perfectness: A subgame perfeet equilibrium point

. . .

b

=

(bi,...,bn) of an extensive game r is an equilibrium point (in behavior strategies) which induces an equilibrium

point on every subgameof

r.

5. A

numerical example

The definition of a subgame perfeet equilibrium point ex- cludes some cases of intu1tively unreasonable equilibrium points for extensive games. In this section we shall present a numerical example which shows that not every intuitively unreasonable equilibrium point is excluded by this defini- tion. The discussion of the example will exhibit the nature of the difficulty.

The numerical example is the game of figure _1. Obviously this game has no subgames. Every player has exactly one information set. The game is agame with perfeet recall.

Since every player has two choices, Land R, a behavior

strategy of player i can be characterized by the probability with which he selects R. The symbol Pi will be used for

this probability. A combinat1on of behavior strategies is represented by a triple (P1,P2,P3).

As the reader can verify for himself without much difficul- ty the game of figure 1 has the following two types of equi- librium points:

Type 1: P1 = 1, P2 = 1, o t:-

-

^{P3 L}

-

^4'1 Type 2: P1 =

0, _-

¹ L.P /, 1

,

^P

-

¹

3- 2 ....

3 -

- 13 -

Consider the equilibriurn points of type 2. Player 2's information set is not reached, if an equilibriurn point of this kind is played. Therefore his expected payoff does not depend on his strategy. This is the reason why his equilibrium strategy is best reply to the equilibriurn strategies of the other players.

o o o

:3

2 2

o o 1

41

~I

Figure 1: A nurnerical exarnple. Information sets are represented by dashed lines. Choices are indi- cated by the letters Land R (standing for "left"

and "right"). Payoff vectors are indicated by colurnn vectors above the corresponding endpoints.

Now suppose that the players believe that a specific type 2 equilibrium point, say (0,1,1) is the rational way to play the game. Is it really reasonable to be-

lieve that player 2 will choose R if he is reached?

If he believes that player 3 will choose R as prescrib- ed by the equilibrium point, then it is better for hirn to select L where he will get 4 instead of R where he will get 1. The same reasoning applies to the ~ther type 2 equilibrium points, too.

Clearly, the type 2 equilibrium points cannot be re- garded as reasonable. Player 2's choices should not be guided by his payoff expectations in the whole game but by his conditional payoff expectations at x3. The payoff expectation in the whole game is computed on the

assumption that player 1's choice is L. At x3 this as- sumption has been shown to be wrong. Player 2 has to assurne that player 1's choice was R.

For every strategy combination (P1,P2,P3) it is possible to compute player 2's conditional payoff expectations for his choices Land R on the assumption that his in- formation set has been reached. The same cannot be done for player 3. Player 3's information set can be reached in two ways. Consider an equilibrium point of type 1, e.g. the equilibrium point (1,1,0). Suppose that (1,1,0) is believed to be the rational way to play the game

and assume that contrary to the expectations generated by this belief, player 3's information set is reached.

In this case player 3 must conclude that either player 1 or player 2 must have deviated from the rational way of playing the game but he does not know which one.

He has no obvious way of cornputinga conditional proba- bility distribution over the vertices in his information set, which teIls hirn,with which probabilities he is at x1 and at x2 if he has to rnakehis choice.

- 15 -

In the next seetion a model will be introdueed whieh is based on the idea that with some very small probability a player will make amistake. These mi stake probabili- ties do not direetly generate a eonditional probability distribution over the vertiee of player 3's information

set. As we shall see in seetion 8 the introduetion of slight mistakes may lead to a strategie situation where the rational strategies add some small voluntary deviations to the mistakes.

6. A model of slight mistakes

There eannot be any mistakes if the players are abso- lutely rational. Nevertheless, a satisfaetory interpre- tation of equilibrium points in extensive games seems' to require that the possibility of mistakes is not

eompletely exeluded. This ean be achieved by a point of view whieh looks at complete rationality as a lirniting ease of ineomplete rationality.

Suppose that the personal players in an extensive game r with perfeet recall are subjeet to a slight imperfeetion of rationality of the following kind. At every information

set u there is a small positive probability Eu for the breakdown of rationality. Whenever rationality breaks down, every choiee c at u will be seleeted with sorne positive probability qc whieh may be thought of as de- termined by sorne unspecified psychological meehanisrn.

Eaeh of the probabilities Eu and qe is assurned to be

independentof all the other ones.

Suppose that the rational choice whieh seleets e with probability bability of the choiee c will be

at u is a local strategy p_c^. Then the total pro-

The introduetion of the probabilities _~EU and qc transforms the original garne into a changed game r where the players do not eompletely eontrol their choiees. A garne of this

kind will be called a perturbed game of r.

Obviously, it is not important whether the Pc or the ßc are consider~d to be the strategic variables of the perturbed game r. In the following we shall take the_A

latter point of view. This means that in r every player i selects a behavior strategy which assigns probability distributions over the choices c at u to the information

sets u of player i in such a way that the probability

p _c

assigned to a choice c at u always satisfies the fol- lowing condition:

(10)

The probability

p

c is also restricted by the upper bound 1-cu(1-qc); it is not necessary to introduce this upper bound explicitly since it is implied by the lower bounds on the probabilities of the other choices at the same information set. With the help of the notation

condition ₍₁₀₎ can be rewritten as follows:

(12) _{for every personal choice c.}

Consider a system of positive constants EC for the perso- nal choices c in r such that

(13)

L ^nc

^{< 1}

c at C u

Obviously for every system of this kind we can determine positive probabilities_A Eu and qc which generate a per- turbed game r whose conditions (10) coincide with the conditions (12). Therefore we may use the following definition of a perturbed game.

A

Definition: A perturbed game r ie a pair (r,n) where r is an extensive game with perfect recall and n is a function

- 17 -

which assigns a positive probability nc to every personal ahoice c in r such that (13) is satisfied.

The probabilities" c are called minimum probabilities.

For every choice c at a personal information set u define (14) ~

=

1 + "

C C ^1"1'C ^,

c'at u

obviously ~c is the upper bound of Pc implied conditions (7). This probability ~c is called probability of c.

by the

the maximum

Strategies: A local strategy for the perturbed game

A

r - (r,n) is a local strategy for r which satisfies _Athe conditions (12). A behavior strategy of player i in r is a behavior strategy_A of player i in r which assigns local

strategies for r to the information sets of pla~er i. The

set of all behavior strategies of player i for_{. .}r is denot- ed by Bi. A behavior strategy combination for r is a be- havior strategy combination D

=

^(D₁^,...,B_{A n}^{) for r whose}

components are behavior strategies for r. The set of all

A A

behavior strategy combinations for r is denoted by B.

Best replies: Let b =(b1,...,b ) be a behavior strategy

A n ~

comb:nation for r. A behavior strateg~ 0i of player i for r is called a best reply to b in r if we have

(15)

~ ~ ~ A

A behavior strategy combination D

=

(D1^{,...,5 )for r is}

. n ~

called a best reply to b.in r if every component 5i of bi is a best reply to b in r.

Equilibrium point: An equilibrium point of_.

strategy co mb in at ion for r wh ich is a best

A

in r.

r is a behavior reply to itself

Remark: Note that there is a difference between a best

A A

reply in rand a best reply in r. The strategy sets Bi are subsets of the st:ategy sets Bi. Pure strategies are not available in r.

7. Perfect equilibrium points

The difficulties which should be avoided by a satisfactory definition of a perfect equilibrium point are connected to unreached information sets. There cannot be any un- reached information sets in the perturbed game. If b is a behavior strategy combination for the perturbed game then the realization probability p(x,b) is positive for every vertex x of K. This makes it advantageous to look

A

at agame r as a limiting case of perturbed games r= (r,n).

In the following a perfect equilibrium point will be de- fined as a limit of equilibrium points for perturbed games.

Sequences of perturbed games: Let r be an extensive game A1 A2

with perfect recall. A sequence r , r ,...

where for

Ak k

k = 1,2,... the game r = (r,n ) is a perturbed game of r, 18 called a test sequence for r, if for every choice c of the personal players in r the sequence of the minimum

k k

probabilities nc assigned to c by n converges to 0 for k +-.

A1 A2

Let

r

^{, r ,} ... be a test sequence for r. A behavior

~

strategy combination b for r is called a limit equilibrium point of this test sequence if for k

=

1,~,... an equili-

Ak Ak

brium point b of r can be found such that for k+- the

Ak .

sequence of the b converges to b .

.

Lemma 3: A limit equilibrium point b of a test sequence

A1 A2

r ^, r ,... for an extensive game r with perfect recall

is an equilibrium point of r.

k

Proof: The fact that the bare equilibrium points of the Ak

r can be expressed by the following inequalities

- 19 -

(16) "'"k

biEBi and

"'"k

Bi with k

for i=1,...,n.

Let

~

be the interseetion of all k~ m we have

~ m. For

Sinee the expeeted payoff depends eontinuously on the be- habior strategy eombination this inequality remains va-

lid if on both sides we take the limits for k+-. This yields:

(18 )

Inequality (18) holds m

of all Bi is Bi. This yields:

for every m. The elosure of the union together with the eontinuity of Hi

(19 )

Inequality (19) shows that b is an equilibrium point of r.

Per feet equilibrium point: Let r be an extensive game with perfeet reeall. A perfeet equilibrium point of r is a be-

R R R

havior strategy eombination b

=

^(b_1,...,bn^{) for}

^r

^{with the}

, "'"1 "'"2

property that for at least one test sequenee r ,

r

J..

...

R 1 "'"2

the eombination b is a limit equilibrium point of r ,r ,...

R

Interpretation: A limit equilibrium point b of a test se- quenee has the property that it is possible to find equili-

*

brium points of perturbed games as elose to b as desired.

The definition of aperfeet equilibrium point is apreeise statement of the intuitive idea that a reasonable equili- briums point should have an interpretation in terms of arbi- trarily small imperfeetions of rationality. A test se-

R

quenee whieh has b as limit equilibrium point provides R

an interpretation of this kind. If b fails to be the limit

*

equilibrium point of at least one test sequenee b must be

regarded as instable against very small deviations from per- feet rationality.

Up to now it has not been shown that perfeetness implies sub- game perfeetnes. In order to do this we need a lemma on the subgame perfectness of equilibrium points for perturbed games.

Subgames of perturbed ~ames: Let r

=

(~,~) be a perturbed

game of r. A subgame r'

=

(f',n') of r consists of a subgame r' of rand the restriction ~' of n to the personal choices

A

of r'. We_A say_A that r' is generated by r'. An equilibrium_.

~oint b of r is called subgame ~erfec~ if an equilibrium point

b' is indueed on every subgamer' of r.

Lemma_~ 3: Let r be an extensive game with perfect reeall and let_A r =

(r,n)

be a perturbed game of r. Every equilibrium point of ^r (in behavior strategies) is subgame perfect.

Proof: Let b' be the behavior strategy combination induced

A A

by an equilibrium_A point b of r on a subgame r' of r. Obvious- ly b' is a behavior strategy combination for the subgame

A A

r' =(r',n') generated _Aby r'. Suppose that b' fails to be an equilibrium point of r'. It follows_. that_A for some personal player j a behavior strat;qy bj forAr' exist, such that player

j's expected payo!f forAb'/bj in r' is greater than his ex- peeted payoff_A for b' in r'. Consider the behavior strategy bj fo~ r wh~ch agrees with bj on r' and with player j's stra- tegy bj in Ab everywhere else. Since the realization probabi- liti=s in rare always positive player j's expected payoff for b/bj must be greater than his expected payoff for b.

Since aAbehavior strategy bj with thi~ property does not eXist, b' is an equilibrium point of r'.

Theorem 2: Let r be an extensive game with perfect recall and let ~ be a perfect equilibrium point of r. On every sub-

~

game r' of r a perfect equilibrium point b' is induced by

~ on r'.

21 - Corollary :

game r with point of r.

Every perfect equilibrim point of an extensive perfect recall is a subgame perfect equilibrium

'"'1 '"'2

Proof: Let r ^{, r} ,... be a test sequence for r which has b as

""" '"'2

limit equilibrium point. Let b ; b ,... be a sequence of equi-

'"'k '"'k

librium points b of r . It follows from the subgame perfectness

'"'k '"'k

of the b that the subqames of r generated by r'form a test

'"

sequence for r' with b' as a limit equilibrium point. Therefore

~

51 is a perfect equilibrium point of r'.

The corollary is an immediate consequence of the fact that a

perfeet equilibrium point is an equilibrium point. (See lemma 3.) 8. A second look at the numerical example

In this section we shall first look at a special test sequence of the numerical example of figure , in order to compute its limit equilibrium point. The way in which this limit equilibrium point is approached exhibits an interesting phenomenon which is important for the interpretation of perfect equilibrium points. Later we shall show that every equilibrium point of type , is perfect.

Let €1'€2"" be a monotonically decreasing sequence of positive probabilities with €_, ^< _,

4 and €k ⁺⁰ for k + -. Let rbe the game

'"'1 '"'2

of figure 1. Consider the following test sequence r ^, r ,... for r.

'"'k

k

For k = 1,2,... the perturbed game r

=

(y,~ ) is defined by k

~c = €k for every choice c of ^r.

As in section 6 let Pi be the probability of player ils choice R.

A behavior strategy combination can be represented ba a triple

'"'k p

=

(P1,P2,P3)' The behavior strategy combinations for rare restricted by the condition

for i = 1,2,3

As we point

'"'k

ahall see, the perturbed game r

k k k k

P = (P1,P2,P3 ) whose components

has only one equilibrium k

Pi are as follows:

k

Equilibrium property of p: In the following it will be shown

k Ak

that p is an equilibrium point of

r

^. Let us first look at the situation of player 3. For any p

=

(P"P2,P3) the reali-

zation probabilities p(x"p) and p(x2,p) of the vertices x, and x2 in the information set of player 3 are given by (24) and (25).

(24) p(x"p)

=

'-p, (25 )

Player 3's expected payoff under the condition that his

information set is reached is 2p(x,p) if he takes his choice R and p(x2,p) ifAhe takes his choice L. Therefore P3 is a best reply to p in rk if and only if the following is true:

(29) (30)

Therefore it follows by (27) that p~ is a best reply to pk.

(21) P1k

=

1 - Ek (22) k _ 1 _ _2Ek

P2 - 1-Ek

k 1

(23) P3 = 4"

(26) P3 = Ek for 2 (1-p,) < P, (1-P2) (27 )

Ek P3 {'-e:k for 2(1-p,)

=

P1(1-P2) (28) P3

=

1-Ek for 2 (1-P1) > P1 (1-P2) In the case of Pk

we have

- 23 -

Let us now look at the situation of player

2.

^{Here we can}

Ak

see that P2 is a best reply to p in r if and only if the following is true:

k k

P2 is best reply to P in view of (32).

Ak

P1 is a best reply to P in r if and only if the following is true:

Uniqueness of the equilibrium point: In the following it

k Ak

will be shown that P is the only equilibrium point of

r .

We first exclude the possibilityP3 ~ 1/4. Suppose that p is an equilibrium point with P3 < 1/4. It follows by

(33) that we have P2

=

^1-e:k. Concequently 3P3 is smaller than P2 and (36) yields P1

=

^{1-e:k. Therefore} (28) applies

to P3. We have P3

=

1-Ek contrary to the assumption P3 < 1/4.

Now we suppose Condition (31) condition (36) to P3 contrary

that P is an equilibrium point with P3 > 1/4.

yields P2

=

^Ek. In view of 1-P2 > 3/4

applies to P1. It follows that (26) applies to the assumption P3 > 1/4.

(31) P2

=

^Ek for P3 > '41

(32 ) Ek P2 1-Ek for P3

=

^4"1

(33 ) P2

=

^{1-Ek for P3 < 1.4}

(34) P1

=

^{€k for 3P3} > 4(1-P2)P3+P2

(35 ) Ek P1 1-e:k for 3P3

=

4 (1-P2)P3+P2

(36) P1

=

1-e:k for 3P3 < 4 (1-P2)P3+P2

k k

(36) ^. P1 is a best reply to p in view of

We know now that an equilibrium point p of Ak

r

must have the 1 Obviously (36) applies to an equilibrium property P3

=

^4".

point p. We must have P1

=

1-e:k ^. Moreover neither ^{(26) nor}

(28) are satisfied by P3. Therefore in view of (27) an equilibrium point p has the following property:

(37 )

This together with P1

=

1-Ek yields 2Ek

P2 = 1-&

,k (38)

equilibrium point:

For k+- the This is the

"'1 sequence r ,

.

converges to p =(1,1,1/4).

equilibrium point of the test

k . k

Note that P1 is as near as possible to P1

=

1 since P1

k is the maximum probability 1 - Ek.

*

Contrary to this P2

*

is not as near as possible to P2. The probability P2 is smaller than 1 - Ek by Ek(1+€k) / (1 - Ek). The rules of the perturbed game force player 2 to take his choice L with a probability of at least Ek but to this minimum probability he adds the "voluntary" probability

€k(1+€k) / (1-€k). In this sense we can speak of a vo- luntary deviation from the limit equilibrium point.

The voluntary deviation influences the realization proba-

k k

bilities p(x1,p ) and p(x2,p ). The conditional probabi- lities for x] and x2,if the information set of player 3 is

R '

reached by p ^, are 1/3 and 2/3 for every k. It is natural to think of these conditional probabilities as conditional

*

probabilities for the limit equilibrium point p ,too, The assumptions on the probabilities of slight mistakes

"'1 "'2

which are embodied in the test sequence r , r ,...

do not

directly determine these conditional probabilities but indirectly via the quilibrium points pk

Perfectness of the equilibrium points of type 1: In the fol- lowing it will be shown that every equilibrium point of

. .

type 1 is perfect. Let p

=

(1,1,P3) be one of these equili-

"'1 '"

brium points. We construct a test sequencer , r2, ...